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Abstract
Bound states in the continuum (BICs) allow to obtain an ultrahigh-quality-factor optical cavity. Nevertheless, BICs must be extended in one or more directions, substantially increasing the device footprint. Although super-cavity mode quasi-BICs supported by single nanopillars have been demonstrated recently, their low-quality factor and localized electromagnetic field inside the dielectric nanopillar are insufficient for high-sensitivity refractive index sensing applications. We propose a ring structure rotated by a dielectric sectorial nanostructure, which can achieve a high quality factor by breaking the rotational symmetry of the ring structure with a footprint as small as 3 µm2. As a straightforward application, we demonstrate high performance local refractive index and nanoscale film thickness sensing based on rotational symmetry breaking induced BICs. These BICs reach quality factor and sensitivity of one order of magnitude better than those of conventional super-cavity mode BICs. The proposed method provides insights into the design of compact high quality factor photonic devices, opening up new possibilities for applications in refractive index and nanoscale film thickness sensing.
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1. Introduction
Optical metasurfaces with high quality factor (Q factor) can substantially increase the time of light–matter interactions and are widely used in lasers [1–3], sensors [4–7], nonlinear effects [8–11], and other fields. Physically, the Q factor (spectral linewidth) of an optical metasurface depends on the absorption and radiation losses of the device. Currently, replacing high-absorption metal structures with low-loss dielectric structures allows to improve the Q factor of metasurfaces [12–18]. However, owing to the unavoidable multichannel radiation loss of the nanocavity, further improving the Q factor is difficult by simply controlling the material absorption. Even when dielectric materials with low absorption loss are used, the Q factor of the high-order Mie resonance is approximately twenty [18]. Hence, further improving the Q factor requires inhibiting the radiation channel [19,20].
In optics, bound states in the continuum (BIC) refer to a phenomenon in which light waves do not produce radiation loss in an open photonic crystal structure [21–29]. This mode is above the light cone of the crystal band. However, owing to the restriction of symmetry or destructive interference, it is completely bound and cannot radiate to the far field. The traditional BIC structure stems from the interaction between periodic lattices and requires an infinite dimension, which can theoretically achieve an infinite Q factor. This periodic structure can be used in applications such as enhanced nonlinear effects, refractive index sensing, and optical trapping. However, periodic BICs are sensitive to a wide range of refractive-index changes and do not allow to detect local changes [30–32].
Recently, aperiodic quasi-BIC devices have been reported, such as super-cavity mode resonators that achieve high Q factors by manipulating the coupling between Fabry–Perot and Mie resonances supported by a single dielectric nanopillar [17,33–35] with a Q factor of approximately 200. In addition, a single GaAsP nanopillar laser based on the super-cavity mode has been realized in a liquid nitrogen environment with an ultralow pumping threshold energy density [36]. Also, electromagnetic field enhancement was achieved using a super-cavity mode supported by a single silicon nanopillar was used to realize amplified spontaneous emission of silicon metasurface under femtosecond laser excitation [37]. Very recently, a quasi-BIC structure based on an aperiodic radially polarized beam was proposed by breaking the radial symmetry of a ring resonator, which has high Q factors with an ultracompact footprint, enabling a high nonlinear coefficient and high sensitivity to the surrounding refractive index [38].
We introduce a high-Q factor azimuthal polarized quasi-BIC with an ultracompact footprint of approximately 3 µm2 by azimuthal symmetry breaking of the ring structures. Compared with super-cavity mode quasi-BICs(SC-BIC), in which the electromagnetic field is localized within the silicon nanopillar, the electromagnetic field of the rotational symmetry-breaking-induced BIC (RS-BIC) diverges to the gap of the silicon nanostructure, strengthening the light–matter interactions. As a proof-of-principle demonstration, we exploit the potential of RS-BIC for local refractive index and nanoscale film thickness sensing.
2. Results and discussion
The SC-BIC of single nanopillar can be efficiently excite by using azimuthally polarized beam [17], as shown in Fig. 1(a), with height h1 = 635 nm and variable radius r1. Owing to the strong coupling between the Mie resonance and Fabry–Perot mode supported in dielectric nanostructures, the SC-BIC is formed, which was applied to obtain high-efficiency nonlinear effects [17]. Although this kind of strong coupling can realize a quasi-BIC, it requires accurately adjustment of the geometrical parameters of the dielectric nanostructures to ensure that the phase and amplitude of the two modes provide completely destructive interference, implying that it is difficult to achieve the highest Q factor in experiment because of manufacturing errors. We obtain a high quality-factor RS-BIC by breaking the rotational symmetry using an azimuthally polarized pump light to excite dielectric nanostructures with a small footprint size(∼3µm2), as illustrated in Fig. 1(b-c). The sectorial structure with central angle β=9° is rotated along the z axis. Two adjacent sectorial nanostructures form angle α. The inner and outer radii of the nanostructure are r2 = 1.13 µm and r3 = 1.77 µm, respectively, and height h2 is 450 nm. Azimuthally polarized light is incident along the z axis to excite the quasi-BIC. We model this system by using finite element method (Multiphysics, COMSOL). The material of nano-structural was set as silicon. To simplify the model, we assume that the permittivity of the nanostructure is a constant of approximately 10.34 within the frequency range of interest.
Fig. 1. Mechanisms of quasi-BIC for azimuthal polarized beam. (a) Super-cavity mode azimuthal polarized quasi-BIC(SC-BIC). (b) and (c) Top view and 3D view of the rotational symmetry breaking induced azimuthal quasi-BIC(RS-BIC) with α = 15°
The transmission spectra of the two types of nanostructures proposed above are shown in Fig. 2. The wave function of azimuthal polarized light transmitted along the z axis is set to ${\vec{E}_i}(\rho ,\varphi ) = {l_0}P(\rho )[\textrm{cos}{\varphi _0}{\vec{e}_\rho } + \textrm{sin}{\varphi _0}{\vec{e}_\varphi }]$, with l0 being the peak field amplitude at the pupil plane, P(ρ) being the axially symmetric pupil plane amplitude distribution. The unit vectors being given by [39]
Fig. 2. Characterization of spectral response of azimuthal polarized BIC. (a-b) Evolution of the transmittance spectra with increasing radius r1 and angle α for SC-BIC (a) and RS-BIC (b). (c) Transmittance spectrum for various values of radius r1 of dielectric nanopillar. (d) Transmittance spectrum for various values of angle of rotation α of dielectric ring structure (α = 18 ° corresponds to the symmetric ring nanostructure). (e) Dependence of Q factor of two kind of BICs on the geometry of nanostructure (i.e., angle α for RS-BIC and radius r1 for SC-BIC).
The amplitude distribution over the pupil is mapped onto the spherical wavefront through ray projection function g(0) given by $\rho /f = g(\theta )$, where f is the objective lens’s focal length and P(0) is the pupil apodization function. The spherical wavefront can be obtained considering power conservation as follows:
Figure 2(a-b) show the evolution of the transmittance spectra with increasing radius r1 and angle α for SC-BIC and RS-BIC, respectively. Note that when the silicon nanopillar is excited using azimuthally polarized beams, we cannot directly observe the previously mentioned two modes in the transmittance spectra of Fig. 2(a) [17]. We further provided the selected transmittance spectra of SC-BIC in Fig. 2(c). For the silicon nanopillar that supports the SC-BIC, a valley appears in the transmitted optical intensity with radius r1 ranging from 405 nm to 495 nm, as shown in Fig. 2(a). With increasing radius, the response spectrum of the nanostructure exhibits two characteristics. First, the valley of the transmitted optical intensity is redshifted from 1500 nm to 1600 nm. Second, the linewidth of the transmission spectrum decreases and then gradually increases. When the radius r1 is 455 nm, the linewidth reaches a minimum, that is, the quasi-BIC condition. We calculate the dependence of the Q factor (Q = λ/FWHM for full width at half maximum FWHM) on the radius as the red-labeled curve shown in Fig. 2(e). The Q factor reaches a maximum of approximately 105 at the quasi-BIC point for r1 = 455 nm. The device is very sensitive to the radius. When r1 deviates by approximately 20 nm from the quasi-BIC point, the Q factor decreases by approximately 50%. Thus, a rigorous procedure is required for experimental preparation.
For comparison, we obtain a high-Q factor quasi-BIC by breaking the rotational symmetry of the ring structure, as illustrated in Fig. 1(b). When the symmetry of the ring structure is maintained (i.e., α = 18°), the ring nanostructure cannot excite a BIC, as shown in the Fig. 2(b). One can clearly see that the transmission spectrum of the ring structure increases linewidth as α decreases. We further provided the selected transmittance spectra of RS-BIC in Fig. 2(d). When α = 17°, the line width is approximately 1.5 nm, corresponding to a Q factor of approximately 1250. This is one order of magnitude greater than the Q factor (approximately 105) of the SC-BIC. Figure 2(e) shows the dependence of the Q factor of the RS-BIC on angle α. When α ranges from 17° to 14°, the Q factor is always greater than 100, reducing the difficulty of experimental preparation. The Q factor of approximately 1250 corresponds to α = 17°, which can be further improved as α approaches 18°. Theoretically, the Q factor is proportional to Q0/(δα)2 [40], where Q0 is a constant determined by structure design and material refractive index and δα = 18 − α represents the perturbation of the rotational symmetry of the nanostructure. The RS-BIC proposed here can be fabricated by the state-of-the-art nano-fabrication technology [38], that is, the nano-patterns defined by electron-beam lithography and transferred into the crystal silicon-on-insulator substrate using reactive-ion etching. Note that the smaller rotational symmetry perturbation of RS-BIC corresponds to a higher Q-factors, reducing the fabrication processing difficulty.
To gain deeper insights into the optical specificity of the two types of BICs, Fig. 3(a) shows their electromagnetic field distributions. The electromagnetic field distribution of the SC-BIC is rotationally symmetric and localized in the interior of the dielectric nanopillar [17]. The electromagnetic field distribution of the RS-BIC diffuses in the gap between the dielectric ring structure, as shown in Fig. 3(b). Compared with their RS-BIC, the spatial profile of the azimuthally polarized vector pump beam can efficiently couple with the distribution of the SC-BIC mode, enabling a strong nonlinear conversion coefficient in experiment [17]. In refractive-index sensing, for the measured external material to interact strongly with the optical mode of the dielectric nanostructure, the electromagnetic field must be localized outside the dielectric nanostructure. Hence, the RS-BIC is more suitable for refractive index sensing than the SC-BIC. To quantitatively describe the optical mode of this type of BIC, we decompose the optical mode supported by the dielectric nanostructure using multipole decomposition, which provides the radiation intensity of dipole moments in the far field from the current inside the silicon nanostructure. The total scattering power can be obtained, including the multipoles: electric dipole (ED), magnetic dipole(MD), electric quadrupole(EQ), magnetic quadrupole (MQ), and toroidal dipole (TD). The calculations are expressed as follows. Electric dipole moment [41]:
Fig. 3. Mode analysis of two types of BICs. (a) Spatial distribution of electromagnetic field and (b-c) multipolar decomposition of RS-BICs and SC-BICs, respectively.
Magnetic dipole moment:
Toroidal dipole moment:
Electric quadrupole moment:
Magnetic quadrupole moment:
Variables ɛ0, ɛp, and ɛd are the permeabilities of vacuum, nanostructure (ɛd = 11.56 in this study), and environment (1 in this study), respectively. The scattering powers of each multipolar component ${\sigma _{ED}},{\sigma _{MD}}$, ${\sigma _{TD}}$, ${\sigma _{EQ}}$, ${\sigma _{MQ}}$ can be readily obtained as $2{\omega ^4}/(3{c^3})|{\textbf p}{|^2}$, $2{\omega ^4}/(3{c^3})|{\textbf m}{|^2}$, $2{\omega ^6}/(3{c^5})|{\textbf T}{|^2}$, ${\omega ^6}/(5{c^5})|eq{|^2}$, ${\omega ^6}/(40{c^5})|mq{|^2}$,respectively. The results of multipole decomposition are shown in Fig. 3(b), where the SC-BIC is dominated by the magnetic dipole mode, whereas the RS-BIC includes the magnetic dipole and magnetic quadrupole modes. Physically, an electric (magnetic) quadrupole supported by a dielectric nanostructure result in an anisotropic electromagnetic field spatial distribution, enhancing light–matter interactions.
To verify the performance of the proposed local refractive-index sensing based on the azimuthal quasi-BIC, we have performed simulation to calculate the evolution of the transmittance spectrum with increasing refractive index n. Here we define α = 13° and α = 15° as RS-BIC1 and RS-BIC2, respectively. Figure 4(a) shows the dependence of the transmittance spectrum of our proposed RS-BIC1 on the refractive index of the surrounding environment. The volume of the simulation region is 3 µm × 3 µm × 5 µm with a perfectly matched boundary condition. We sweep the refractive index of the entire simulation region for n ranging from 1 to 1.1(corresponding to the gaseous materials [42,43]). The transmission peak of the RS-BIC moves from 1550 nm to 1590 nm, corresponding to the SC-BIC moving only approximately 14 nm, as shown in Fig. 4(b). We define the refractive-index sensitivity, S, and Q factor figure of merit, FOM, for our refractive-index sensor as follows:
Fig. 4. Dependence of spectral response of (a) RS-BIC1, (b) SC-BIC and (c) RS-BIC2 on refractive index of surrounding environment. (d) Statistical distribution of Sn and FOMn of RS-BIC1, RS-BIC2 and SC-BIC.
Both Sn and FOMn of the RS-BIC1 are 312 nm/RIU and 30, respectively, corresponding of the SC-BIC are only 128 nm/RIU and 6.5, respectively. In addition, due to the FOMn = Sn⁄FWHM is inversely proportional to the full-width-at-half-maximum of transmittance spectrum, it can be further improved by reducing the angle perturbation of the RS-BIC (i.e., α close to 18°), as shown in Fig. 4(c) (RS-BIC2). Owing to the electric field of the RS-BIC extend to outside of the ring structure, it efficiently interacts with the external material inside the gap of the dielectric ring structure, enabling the Sn and FOMn of the RS-BIC2 are 3 and 10 times larger than those of the SC-BIC, respectively. Fig.4d shows the statistical results of performance parameters for RS-BIC1, RS-BIC2 and SC-BIC, respectively. The asymmetric value of angle α is 15°, a smaller line width can be achieved by letting the rotation angle to approach 18°, which can further increase FOMn.
In addition, we demonstrate the application of the two types of azimuthal polarized quasi-BICs to nanoscale thin-film thicknesses sensing. The transmittance spectra of RS-BIC1, covered with different thicknesses of conformal silica (n = 1.45) thin films. The dependence of the transmittance spectra of the two types of BICs on the thickness (s) of the silica layer for RS-BIC1 and SC-BIC is shown in Figs. 5(a) and 5(b). With increasing thickness t, the peak of the RS-BIC gradually redshifts from 1545 nm to 1640 nm, while the peak of the SC-BIC redshifts from 1555 nm to 1610 nm. We define sensitivity parameters St = Δλ⁄Δt and FOMt = St⁄FWHM to characterize the sensitivity of the system for the thin-film thickness sensing. St and FOMt of the RS-BIC are 4.6 and 0.29, corresponding to the SC- BIC 2.9 and 0.1, respectively. Remarkably, the FOMt can reach up to 1 by increasing the Q factor of the nanostructure, i.e., RS-BIC2, as shown in Fig.5c. Fig.5d shows the statistical results of film thicknesses sensing performance for RS-BIC1, RS-BIC2 and SC-BIC, respectively. In principle, the drastic difference in the sensitivity of thin-film sensing arises from the spatial distribution of the electromagnetic fields.
Fig. 5. Dependence of spectral response of (a) RS-BIC1, (b) SC-BIC and (c) RS-BIC2 on refractive index of surrounding environment. (d) Statistical distribution of St and FOMt of RS-BIC1, RS-BIC2 and SC-BIC.
3. Conclusion
We theoretically analyze azimuthal polarized quasi-BICs and demonstrate their capability for sensing the local refractive index and nanoscale thin-film thickness. The RS-BIC has a high Q factor up to 1500, which is one order of magnitude higher than that of the SC- BIC. The RS-BIC is dominated by a magnetic dipole and magnetic quadrupole with an extended electromagnetic field in the gap of the dielectric ring structure, whereas the SC-BIC is localized inside the dielectric nanopillar. We demonstrate that it has a high capability for local refractive index sensing and nanoscale film thickness sensing. The FOMn and FOMt of the RS-BIC can reach 60 and 1, respectively, being one order of magnitude higher than the SC-BIC. The proposed azimuthal polarized RS-BIC may be applied to sensing at the cellular and molecular scales in future work.
Funding
Natural Science Foundation of Chongqing (2023NSCQ-MSX2201); National Natural Science Foundation of China (62305035).
Acknowledgments
The authors are grateful to OSA Publishing Language Editing Services for its linguistic assistance during the preparation of this article.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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